ICS 252 Introduction to Computer Design Lecture 9 Winter 2004 Eli Bozorgzadeh Computer Science Department-UCI

2 Winter 2004ICS 252-Intro to Computer Design Reference –Lecture note by Rajesh Gupta on resource sharing [©Gupta] (lecture 8) –Chapter 7 of the textbook Administrative –Midterm 1: Feb 13, 10 a.m. in the class –Will you to meet me for your final project early next week

3 Winter 2004ICS 252-Intro to Computer Design Logic Optimization Logic synthesis synonymous with logic optimization Optimization depending on Implementation style –Two-level (PLA) –Multi-level (cell- or array-based) Circuit operation –Combinational –Sequential

4 Winter 2004ICS 252-Intro to Computer Design Agenda Combinational logic design basics Exact optimization for two-level logic Heuristic optimization for two-level logic Symbolic optimization and encoding problems

5 Winter 2004ICS 252-Intro to Computer Design Combinational logic design basics Two-level representation Cubes and covers –Cover C of f if f ON  C  f ON Uf DC Minimum cover –No cover with fewer cubes Irredundant cover –If no subset of C is a cover of f –No implicant can be dropped Minimal cover w.r.t. single-cube containment (SCC) –No cube of C is contained in another cube of C Irredundant  SCC (is it true for converse?

6 Winter 2004ICS 252-Intro to Computer Design Cont… Prime implicants –Prime implicant not contained by any other implicant –A product of literals where no literal can be dropped –Geometrically : largest size cube without intersecting the OFF-SET Essential prime implicant –Must be contained in any cover of the function

7 Winter 2004ICS 252-Intro to Computer Design Exact optimization for two-level logic Goals: –Reduce number of implicants –Reduce number of literals Determine minimum cover of f Quine: –There exists a minimum cover that is prime Look just for prime implicants Quine-McCluskey method: –Compute prime implicants –Determine minimum cover

8 Winter 2004ICS 252-Intro to Computer Design Prime implicant table A binary-valued matrix A Columns are prime implicants Rows are minterms in the ON-SET a ij =1 iff prime cover j covers minterm I Example: –f=a’b’c’+a’b’c+ab’c+abc+abc’ –Primes? –Implicant table?

9 Winter 2004ICS 252-Intro to Computer Design Cover A Cover can be represented by a (hyper)graph with minterms as vertices and primes as edges (hyper) a b c

10 Winter 2004ICS 252-Intro to Computer Design Covering problem Given A, cover all the rows with least number of columns A minimum cover is a minimum set of columns which covers all rows. Determine x such that Ax  1 Minimize cardinality of x A can be seen as incidence matrix of a hypergraph Column covering as an edge covering problem

11 Winter 2004ICS 252-Intro to Computer Design Covering Brute force : consider all possible values Complexity? Use pruning: Petrick’s method

12 Winter 2004ICS 252-Intro to Computer Design Petrick’s method Write covering clauses of the reduced table in POS form Multiply out POS form into SOP form Select cube of minimum size

13 Winter 2004ICS 252-Intro to Computer Design Example POS form –p 1 (p 1 +p 2 )(p 2 +p 3 )(p 3 +p 4 )p 4 =1 SOP form –p 1 p 2 p 4 +p 1 p 3 p 4 =1 Two minimum covers of cardinality 3 Solution {p 1,p 2,p 4 }, {p 1,p 3,p 4 }

14 Winter 2004ICS 252-Intro to Computer Design Heuristic Method Local minimum cover –Given an initial cover –Make it prime –Make it irredundant Iterative improvement –Reduce cover cardinality by modifying implicants.